Estimation Of Generalized Mixtures And Its Application ... - IEEE Xplore

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DELIGNON et al.: ESTIMATION OF GENERALIZED MIXTURES 1367with(18)(19)Fig. 1.The eight families of Pearson’s system function of (1; 2):in Section III-B, where the shapes of the eight families arerecalled).Let us consider an i.i.d. sequence of real random variableswhose distribution belongs to Pearson’s system.We now specify the estimator used in step 2 of GSEM (seeSection II-B).1) Consider the partition of2) For each class use in order to estimate byParameters are called form parameters. cantake five different forms according to To be more precise1) for density is bell shaped;2) for density is shaped with;3) for density is shaped with;4) for density is shaped with;5) for density is uniform.(Type II Distributions): These distributions are particularcases of obtained for in (17), as follows:withforotherwise(20)(15)(21)(22)for (16)(Gamma Distributions):Densities are given by3) For each class calculate fromaccording to (9).4) For each class use and (14) to estimate whichfamily among the density belongs to.5) With the estimated family and the computed[(10)–(13)], estimate the parameters of the distribution.(For eachthe exact relationship betweendensity parameters and the computedisgiven in the next section.)with(Type IV Distributions):forDensities are given byotherwise(23)(24)B. Shape of Pearson’s System DensitiesIn this section, we specify the shape of the eight distributionfamilies forming Pearson’s system.(Beta Distributions of the First Kind): Densities aregiven byforotherwise(17)with such that and(25)
1368 <strong>IEEE</strong> TRANSACTIONS ON IMAGE PROCESSING, VOL. 6, NO. 10, OCTOBER 1997Fig. 2.Ring image, its noisy version, and results of unsupervised segmentations based on GGICE, GGEM, ASEM, and GSEM.(Inverse Gamma Distributions):Densities are given by(Gaussian Distributions): Densities are given bywithforotherwise(26)(Beta Distributions of the Second Kind): Densitiesgiven bywithforotherwiseare(27)withand(29)C. <strong>Generalized</strong> EM and ICE AlgorithmsThe EM and ICE algorithms are two other mixture estimationmethods that can also be “generalized” to give the GEMand GICE. We briefly describe below their operation.1) GEM: Let be the distributionscomputed fromthe current parameter Priors are reestimated by formula(30), which is the same as that in the EM algorithm, andthe recognition is the same as that the recognitiondescribed at the end of Section III-A, with the difference thatgiven forby formulas(31) and (32), are used instead of those given by formulas(15) and (16).(30)(31)where is the scale parameter and are the formparameters.(Type VII Distributions): Densities are given by(32)withsuch that(28)2) GICE: In the context of this paper, the GICE used isa “mixture” of GSEM and GEM. In fact, the reestimation ofpriors is the same as in GEM, and the family recognition andnoise parameter reestimation is the same as in GSEM.
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